The integration of fins as extended surfaces in phase change material (PCM) heat storage units is receiving significant attention for enhancing phase change effectiveness of PCM in those regions of heat storage unit where minimal phase change of PCM takes place. This enables the uniform melting or solidification of PCM, thereby enhancing the overall performance of heat storage units. Fins of curved shape not only serve as greater extended surfaces but also significantly influence PCM flow dynamics. This paper provides a comprehensive review of recent studies employing various curved fin shapes, such as arc-shaped, wavy-shaped, and other non-conventional forms, in PCM heat storage units, and found that curved fin shapes have a significant impact on PCM flow dynamics, thermal conductivity, and overall unit efficiency. The review includes detailed explanations of the impacts of individual curved fin parameters, such as design, angle, placement, curvature, and optimization techniques. Additionally, the basic operating principles of curved fins in heat storage units are discussed, including the effects of natural convection, numerical model assumptions, and future research directions in this field.